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Financing a new car: You are buying a new car, and you plan to finance your purchase with a loan you will repay over 48 months. The car dealer offers two options: either dealer financing with a low APR, or a \(\$ 2000\) rebate on the purchase price. If you use dealer financing, you will borrow \(\$ 14,000\) at an APR of \(3.9 \%\). If you take the rebate, you will reduce the amount you borrow to \(\$ 12,000\), but you will have to go to the local bank for a loan at an APR of \(8.85 \%\). Should you take the dealer financing or the rebate? How much will you save over the life of the loan by taking the option you chose? To answer the first question, you may need the formula $$ M=\frac{\operatorname{Pr}(1+r)^{48}}{(1+r)^{48}-1} $$ Here \(M\) is your monthly payment, in dollars, if you borrow \(P\) dollars with a term of 48 months at a monthly interest rate of \(r\) (as a decimal), and \(r=\frac{A P R}{12}\).

Step by step solution

01

Calculate Monthly Rate and Payment (Dealer Financing)

First, determine the monthly interest rate for dealer financing. Since the APR is 3.9%, the monthly rate \( r \) is \( \frac{3.9}{100 \times 12} \approx 0.00325 \). Now, compute the monthly payment \( M \) by substituting into the formula:\[ M = \frac{14000 \times 0.00325 \times (1 + 0.00325)^{48}}{(1 + 0.00325)^{48} - 1} \] Calculate the exponentiated terms:\[ (1 + 0.00325)^{48} \approx 1.17072 \] Now substitute back to find:\[ M = \frac{14000 \times 0.00325 \times 1.17072}{0.17072} \approx 314.68 \].Thus, the monthly payment for dealer financing is approximately \$314.68.

02

Calculate Monthly Rate and Payment (Rebate with Bank Loan)

For the bank loan, first determine the monthly interest rate. The APR is 8.85%, so the monthly rate \( r \) is \( \frac{8.85}{100 \times 12} \approx 0.007375 \). Using the formula for \( M \):\[ M = \frac{12000 \times 0.007375 \times (1 + 0.007375)^{48}}{(1 + 0.007375)^{48} - 1} \]Calculate the exponentiated terms:\[ (1 + 0.007375)^{48} \approx 1.432364 \]Now substitute back to find:\[ M = \frac{12000 \times 0.007375 \times 1.432364}{0.432364} \approx 296.00 \].Thus, the monthly payment with the bank loan is approximately \$296.00.

03

Calculate Total Payment and Compare

Now calculate the total payment over 48 months for both options. For dealer financing, the total payment is:\[ 314.68 \times 48 = 15104.64 \] For the bank loan with the rebate, the total payment is:\[ 296.00 \times 48 = 14208.00 \]Now compare the total payments. The bank loan option with the rebate results in a lower total payment compared to dealer financing.

04

Conclusion: Best Financial Option and Savings Calculation

Since the total payment with the bank loan and rebate is \\(14,208.00, which is lower than the dealer's financing total payment of \\)15,104.64, the rebate option is better. Calculate the savings over the 48-month loan:\[ 15104.64 - 14208.00 = 896.64 \]So, by taking the rebate and the bank loan, you save approximately \$896.64.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

APR

APR stands for Annual Percentage Rate and is a crucial term when discussing car loans. It represents the total cost of borrowing over a year, expressed as a percentage. This rate includes any interest you pay on the loan and any fees or additional costs associated with borrowing.

Understanding APR is critical because it provides a basis for comparison between different loan offers. Even if one loan has a lower interest rate, hidden fees can make it more expensive overall. When the APR is lower, it generally means the cost of borrowing is less, leading to potential savings over the life of your loan. In our car loan scenario, comparing the APRs from the dealer and the bank helps us determine which option is more cost-effective.

• A lower APR is beneficial, reducing the total amount you end up paying.
• Always consider the APR when comparing loans, not just the interest rate.
• APR includes more than just the interest rate, giving a comprehensive picture of costs.

monthly payment formula

When you're financing a car, knowing how to compute your monthly payment can be invaluable. The formula given, \[ M = \frac{P \times r \times (1+r)^{48}}{(1+r)^{48} - 1} \] allows us to calculate the monthly payment.

Here, \( M \) is the monthly payment, \( P \) is the loan principal, and \( r \) is the monthly interest rate, which you get by dividing the APR by 12. This formula considers the compound interest effect over the loan period, making it a precise tool for understanding actual monthly obligations.

Remember:

• The higher the interest rate, the higher \( M \) will be.
• A longer loan term lowers monthly payments but increases total interest paid.
• Knowing how to use this formula helps in making informed financial decisions.

interest rates comparison

Comparing interest rates is vital when choosing a loan. Although APR offers an overall picture, interest rate comparison provides immediate insights into day-to-day costs. It influences how much money goes towards the principal versus interest each month.

In our exercise, the dealer offers a lower APR of 3.9% compared to the bank's 8.85%. However, it’s not just about percentages. You need to know how these rates translate into monthly payments and total loan costs over time.

Effective comparison involves:

• Calculating both options' total cost over the loan life.
• Understanding how much goes towards reducing the principal versus interest.
• Analyzing whether additional benefits (like rebates) offset higher rates.

Being thorough in this analysis prevents surprises and helps select the most cost-effective option.

financial decision-making

Making sound financial decisions means analyzing both immediate and long-term impacts of your choices. For our car finance situation, this involves evaluating total costs and potential savings associated with each loan option.

It's essential to ask yourself, "What will save me more money over time?" and "Which option gives me more value for my money?" Our example showed that even with a lower APR, the bank loan with a rebate offered better savings because the total repayment was smaller.

To enhance decision-making, one should:

• Consider total loan costs and interest paid, not just monthly payments.
• Factor in rebates or additional offers which can reduce the principal.
• Use precise calculations and comparisons to ensure optimal financial outcomes.

With informed choices, financial decision-making becomes a powerful tool in managing personal finances effectively.